Question

Let $A=\{1,2\},B=\{1,2,3,4\}$ and $C=\{5,6\}$. Verify that $A\times (B\cap C)=(A\times B)\cap (A\times C)$

Answer 1

$\begin{array}{c}A=\left\{1,2\right\}B=\left\{1,2,3,4\right\}\&C=\left\{5,6\right\}\\ A\times \left(B\cap C\right)=\left\{1,2\right\}\left\{1,2,3,4,5,6\right\}\\ Now,\\ \left(A\times B\right)\cap \left(A\times C\right)=\left\{\left(1,1\right),\left(1,2\right),\left(1,3\right),\left(1,4\right),\left(2,1\right),\left(2,2\right),\left(2,3\right),\left(2,4\right)\right\}\times \left\{\left(1,5\right),\left(1,6\right),\left(2,5\right),\left(2,6\right)\right\}\\ =\left\{1,2,3,4,5,6\right\}\\ \therefore A\times \left(B\cap C\right)=\left(A\times B\right)\cap \left(A\times C\right)\end{array}$